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JOURNAL OF RESEARCH of the National Bureau of Standards— D. Radio Propagation 
Vol. 65D, No. 6, November-December 1961 

Dipole Radiation in a Conducting Half Space 

R. K. Moore and W. E. Blair 

Contribution from University of New Mexico, Albuquerque, N. Mex. 
(Received August 31, 1960; as revised May 9, 1961) 

The problem of communication between antennas, submerged in a conducting medium 
such as sea water, is analyzed in terms of a dipole radiating in a conducting half space sepa- 
rated by a plane boundary from a dielectric half space. The theory is discussed for both 
horizontal and vertical, electric and magnetic dipoles. 

Expressions for the Hertzian potentials of the dipole in the conducting half space can 
be reduced to integrals obtained by Sommerfeld (for a dipole at the boundary) multiplied 
by an exponential depth attenuation factor. The Hertzian potentials are used to determine 
the electric and magnetic field components. 

This analysis shows that the main path of communication between submerged antennas 
is composed of three parts as follows: (a) energy flow from the transmitting dipole directly 
to the surface of the sea, (b) creation of a wave that travels along the surface refracting 
back into the sea, (c) energy flow normal to the surface to the receiving dipole. 

1. Introduction 

Radiation from electric and magnetic dipoles in the air above a plane earth has been con- 
sidered by many writers beginning with Sommerfeld [1909]. Sommerf eld's original paper 
treated a vertical monopole placed at the surface of a flat earth having arbitrary dielectric 
and conductive properties. The solution of the boundary-value problem was based on the 
evaluation of Fourier-Bessel integrals which were solutions to the wave equation. Weyl 
[1919] attacked the problem by resolving the dipole radiation into a spectrum of plane waves 
and evaluating the resulting integral without resorting to Sommerf eld's cylindrical wave 
formulation. Sommerfeld [1926] expanded his original work to take into account vertical 
and horizontal, electric and magnetic dipoles above a plane earth. 

Sommerf eld's original paper contained an error in sign which was a subject of much later 
discussion [Norton, 1935]. Although Sommerfeld's work is followed here, the effect of the 
error in sign does not appear because of the assumption in this paper that the transmitting 
antenna is located in a highly conducting half space. 

In general, papers published prior to 1940 on the subject of radiation of dipoles were 
concerned with dipoles in air or on the surface of the earth. Moreover, several papers have 
been written on this subject since 1940. Study of electromagnetic radiation in the sea began 
about the turn of the century and limited experimental work was done around the end of the 
first world war. Tai [1947] analyzed an electric dipole in an infinite, conducting medium and 
found extremely high dissipation in the vicinity of the dipole because of high conductive losses. 

The first extensive theoretical treatment of electromagnetic radiation by vertical and 
horizontal, electric and magnetic dipoles immersed in a conducting half space appeared in 
a thesis by Moore [1951]. This paper describes a portion of that work. Wait [1952] [1957] 
considered an insulated magnetic dipole in an infinitely conducting medium, showing that the 
fields are independent of the characteristics of the insulation for an antenna diameter much 
less than the radiation wavelength in the conducting medium. Then Wait and Campbell 
[1953] and Wait [1959] analyzed a magnetic dipole of this type in a semi-infinite medium in- 
cluding special cases of frequency, antenna depth, and separation between antennas. Their 
model was a horizontal magnetic dipole (axis parallel to the surface of the sea), submerged 
in the sea (a conducting half space). 

Analysis of the electric dipole was done by Lien and Wait [1953] in which the evaluation 
of the complex integrals was reduced to forms suitable for the numerical computation. Bafios 
and Wesley [1953, 1954] carried out a mathematical analysis of the general case of a horizontal 
electric dipole in a conducting half space. The results in this paper agree with those of Bafios 


to the first order approximation. Kraichman [1960] experimentally verified Wait's and Bafios' 
results in the intermediate distance region range. Also, the exponential increase of attenuation 
with depth was experimentally verified by Saran and Held [I960]. Anderson, in a thesis [1961], 
described the three-layered problem of earth-air-ionosphere for an electric dipole submerged 
in the sea. Further work by Wait [1960a, 1960b] has recently been carried out in an effort to 
unify the various approaches to the problem. 

It is concluded from our analysis that the path of electromagnetic energy, between trans- 
mitting and receiving dipoles in the sea, is the following: (a) propagation from the transmitter 
directly to the surface, (b) propagation along the surface of the sea allowing refraction of the 
energy back into the sea, (c) leakage directly downward into the sea to the receiver. 

A fundamental assumption in this analysis is that the displacement current (in the sea) 
is negligible compared to the conduction current. Since the conductivity of the sea is about 
4 mhos/m and the frequencies of practical use in undersea communication are less than 50 
kc/s, this is not at all a severe restriction for the case considered. Application to related 
problems depends upon the appropriateness of this assumption for each problem. The only 
other restriction is that the energy traveling directly through the sea between the transmitter 
and receiver is neglected. The ratio of the magnitude of the direct wave (through the sea) 
to the over-the-surface wave is of the order of n e~ {R ~ z)/8 where R, z, 5, and n, are respectively 
antenna separation, antenna depth, skin depth, and index of refraction of the sea with respect 
to the air. Thus for Ry>^>z } the ratio is considerably less than unity. 

Because of the u over-the-surface" mode of communication, the really important direction 
for transmitting dipole radiation is directly toward the surface of the sea. Since the radiation 
pattern of the vertical dipoles has a null in this direction, the vertical dipoles are not as effective 
in launching the required wave as are the horizontal dipoles. Electric and magnetic fields have 
been calculated for both the vertical and the horizontal dipoles, but the latter are of much 
greater importance in submarine communications. 

2. Waves in Conducting Media 

Discussions of the nature of electromagnetic radiation in a conducting medium and a 
conducting half space are presented by Stratton [1941] and Sunde [1949]. However, a brief 
review of the pertinent parts of the theory is included here for the purpose of defining the 
variables employed throughout this paper, as well as stating the fundamental conditions of the 
problem of dipole radiation in a conducting half space. 

The Maxwellian curl equations for a conducting medium in which displacement current 
can be neglected are 

VXE=-ju M H (1) 

VXH = ctE (2) 

where the time variation e^ 1 has been suppressed 
E is the electric field vector (volts/meter) 
H is the magnetic field vector (ampere-turns/meter) 
to is the angular frequency (radians/second) 
fx is the magnetic permeability (henrys/meter) 
a is the conductivity (mhos/meter). 

The diffusion equation for E is obtained by combining (1) and (2): 

V 2 E=jojfxaE. (3) 

The solution of (3) for a plane wave traveling in the positive 2-direction is: 

E=E e- {1 + jU/5 with R=e- j ^^2 rk ^- 



where <5 — V2/co/xcr is the skin depth (meters) 

n is the unit vector in the direction of propagation. 
It is convenient in determining the fields of a dipole to determine first either the vector 
potential A or the Hertzian potential II. In this paper, the Hertzian potentials are used for 
both the electric and magnetic dipoles. The electric and magnetic field vectors are functions 
of the Hertzian potential as follows: 

Electric dipole Magnetic dipole 

H^o-VXIl! E^-jcoAtVXn* (4) 

Ei=VV • Hi+AfHi Hi=VV . nf +klnf (5) 

where II, , n* are the Hertzian potentials for the electric and magnetic dipoles respectively, and 

k\ = -y/'—jcofjLo- (meters -1 ) . 

The subscript 1 denotes a conducting medium. Throughout this paper the expressions re- 
lating to the electric dipole are presented in the left column while those for the magnetic dipole 
are in the right column, side by side, for convenient comparison. The Hertzian potentials 
II and II* (the Green's function for a dipole source) in the infinite, conducting medium are 

n ' =Pl ~1T n * =p * ~~R~ (6) 


71 s „ NJS ._, 

Pl =4^ P *=^T (7) 

where R is the distance from the dipole to the observation point 
71=lim (71), is the electric moment (ampere-meters) 



iV/S=lim (iVTS) is the magnetic moment (ampere-meters 2 ) 

I is the current (amperes) . 
1 is the electric dipole length vector (meters) . 
N is the number of turns in the magnetic loop. 
S is the loop area vector (meters 2 ) . 

The corresponding statements for the electric and magnetic fields in an infinite, noncon- 
ducting medium are the following: 

H 2 =jcoe VXII 2 E 2 =-ja>fiVXUt (8) 

E 2 = vv . n 2 +&in 2 H 2 = vv . nf+^n* (9) 

where & 2 =27r/X , is the wave number (meters -1 ) 
X is free-space-wavelength (meters) 
€ is free space permittivity (farads/meter) . 


The Hertzian potentials in this case are 

n 2 = P2 R 

n 2 =pf R 

. II 

P2= 3 A 

P* -' a„ • 



The above statement for the fields in an infinite medium must be modified for the case of 
a source in a conducting half space separated by a plane boundary from a nonconducting 
half space. For the modification, the source (transmitting dipole) is located in the conducting 
half space at coordinate position (0, 0, z t ) from the boundary. Similarly, the point of obser- 
vation (receiving dipole) is located at coordinate position (r, <£, z r ) (see fig. 1). The conducting 
half space (sea) is medium 1 (subscript 1), and the nonconducting half space (air) is medium 
2 (subscript 2). 

The Hertzian potentials in the following are normalized for convenience such that |pi| = 
|p 2 | = |p*| = |p*| = l. The Hertzian potentials in both media must satisfy the wave equa- 
tion (condition I below) and the radiation condition at infinity (condition II) below: 

(I) v»ni+fcjni=o v 2 n*+&?n*=o, Zr yo (sea) 

v 2 n 2 +& 2 n 2 =o v 2 n*+& 2 n*=o, z r <o (air) 

(II) limlli=0 limll*=0 , z r >0 
limll 2 =0 limll*=0 , z r <Q 

where R = ^r 2 +(z t —z r ) 2 . 

Since the source is in the conducting half space, it is also necessary that II x satisfy the radi- 
ation condition near the source : 

(III) limn 1 =^-=UmII?. 

In addition to the above three conditions, the electric and magnetic fields must satisfy the 
boundary conditions at the surface of the sea (the plane, 2=0). Thus the boundary conditions 
for the components of the Hertzian potentials and their derivatives are the following : * 

(IV) - i^n 2l =n 22 ir^n*, 

- jgn xl =ii x2 —iflfE5i=nj a 

du xX _dn x2 dn* xl _dn* x2 


dz dz dz dz 

du xl diL zl= bn x2 m z2 dn* xl dn? 1= dn* 2 an? 2 

dx dz dx dz dx dz dx dz 

where ^r = cr/co^> = | ifci/A:! | • In condition (IV), it is assumed that the dipole is oriented either 
vertically in the ^-direction, or horizontally in ^-direction. (See fig. 1.) Sommerfeld [1949] 
has shown that only a U z component of the Hertzian potential is required to describe the fields 
of a vertical dipole. However, he has shown that both U x and U g components are required 
to describe the fields of a horizontal dipole (oriented in the ^-direction). 

i In condition (IV) the quantity — jg is an approximation for the complex index of reflection squared, n 2 . If the frequency or conductivity 
is such that n* 9*—jg, then one can substitute the exact expression n 2 = —jg+K, where k is the relative permittivity of the medium, throughout this 





Figure 1. Distances and coordinate system. 

3. Hertzian Potential Integrals 

The Hertzian potential can be obtained in integral form for the four basic dipole con- 
figurations: vertical and horizontal, electric and magnetic dipoles. The method used here 
for obtaining the potential integrals was first used by Sommerfeld [1909, 1926]. The method 
is straightforward and available [Stratton, 1941; Sommerfeld 1949], so it will not be presented 
here. A general Hertzian potential for each of the four dipole configurations is obtained 
satisfying conditions (I) through (III). The general solution is then substituted into condition 
(IV) allowing the boundary conditions to determine the arbitrary coefficients of the general 

The Hertzian potentials for the observation point in the sea are presented in integral 
form for the four basic dipole configurations. The superscripts v and h denote vertical and 
horizontal dipole respectively. For z t y>0, z r ^>0, 




COS ip- 


e -jk l R l g--:%# 2 






and where 

£Wr 2 H-(2 r -z,) 2 
R 2 =Tjr*+(z r +z t )* 


lal Jo F-jgG 

Ibl - 2 Jo ~r 

Id=2(-jg-l)J <j 



(F+G) (F-jgG) 
(meters -1 ) 



(meters '). 

The function 0t has two parts: exp (—jkiR^/Ri representing the primary source at the 
position (0, 0, z t ), and exp (—jk 1 R 2 )/R 2 representing a secondary source at the image position 
(0, 0, — z t ). In the case of the horizontal electric and the vertical magnetic antennas, the 
secondary source represents the image of the primary source. However, in the case of the 
vertical electric and the horizontal magnetic antennas, the secondary source represents an 
inverse image — that is, an image dipole of the opposite polarity. The primary source radiates 
over the direct path R 1} and the secondary source radiates over the reflected or image path R 2 
(see fig. 1). Both paths R x and R 2 are entirely within the sea. Consequently, each of the 
Hertzian potentials in (12) and (13) is composed of three components: (a) a primary source 
function, (b) a secondary source function, and (c) an integral. It is shown later that the integral 
in every case represents the major contribution to the Hertzian potentials if r»(2 r +^). 
Therefore it is tacitly assumed that r M, composed of the primary and secondary sources, can be 
neglected. Consequently, (12) and (13) can be approximated for z r ^>0, z t ^>0, r^>^>(z r -\-z t ), 
as follows: 

nW* n*»~/ w (16) 

n2i~/ w n»«J*. (17) 

I al , and I M , and I c i are interdependent. It can be shown that 2 for z^> 0, z r ^> 0, 

^-=I bl +J9hi (18a) 

d r 2 «-*«» ,/i , i\ r q id/., , 1fiM 

Icl ~^rL~M~^^ + \M + M) lal rM~d^' (18b) 

The Hertzian potentials for an observation point located in medium 2 are obtained in the 
same manner as were the Hertzian potentials for a point in medium 1. In the case of the 
potentials in medium 2, the function 3fc is not involved, so the potentials can be written di- 
rectly for z t y> 0, 2 r < 0, 

n* 2 =-j0/ a2 n*f=/ &2 (19) 


Tl h x2 =-J9h2 Ur 2 = -jgla2 (20) 


n? 2 = -jg cos v ^ n* 2 »=cos <p ^f (21) 

Ia2 ~ 2 Jo ~^=WG (22a) 

/,=2 J/ <-* + %fM (22b) 

!*-*{ W i;j o {F+G){G _ jgG) (22c) 
as in (18) , the integrals I a2 , I b2 , and I c2 are also interdependent. For z^> 0, z r <^ 0, 

^=/ S2 -/ a2 (23a) 

2 (18) requires that the term (—jg—1) in (15c) be kept in this form. However, for any practical calculation (— jg—l)~—jg since £» 1 for the 
conductivity and frequencies of interest in this paper. 


Now it is possible 4 to write the electric and magnetic fields in medium 1 (sea) as a function 
of integrals I a \ and I bi ; and the fields in medium 2 (air) as functions of I a2 and I b2 . Since it 
is the purpose of this paper to determine the fields in the sea and at the surface of the sea, 
attention is focused on the integrals I ai and I bi . If the fields in the sea are determined, the 
boundary conditions can be used conveniently to determine the fields in air at the boundary 
(z r = 0—). Consequently, all fields to be discussed in this paper can be expressed in terms 
of integrals I al and I bi . It is convenient, however, to replace I a \ and I bX with two new integrals 
I a and I b with a corresponding change of variables as follows : 



-T - 

1 al- 

=/« = 



C J - 


= C -F 



- Li Jo( P +)W 



= vV 2 + 



= C -G 

O h 

=^'¥ Z 






=- r= 


X /27T 



= — Z- 


= V2x 

z=z r +z t (meters) 

c=speed of light in free space (meters/second) 
X = free-space wave length (meters). 

The dimensionless distance factors p and f are used to express distances r and z in terms of 
(free-space-wavelength) /27T. Therefore the electric and magnetic field components in the sea, 
as functions of the integrals I a and I b are tabulated as follows: 

Electric dipole Magnetic dipole 

Vertical dipole 

Er ~7d^ Wr_ ^^df (26) 

E r =H r =H z =0 H r =E T =E l =0 (29) 

Horizontal dipole 

„ co 2 / d 3 / 8 , 57,\ _, -co 3 /a 3 /, a/A 

ff,= ? c cos „ (^:+ ^r) ff,= - , ^ M cos ^ (^f +^J (31) 


zj <° 2 • <*/» „ .to 3 . bl a . 

H t =—^ ffsm<p-^ E z =j -^ ix sin ^ -^ (32) 

£ r =-^ cos * ^ (!£*+/») ^= "jf cos „ ;V (fjj+J.) (33) 

*,=£ sin „ j9 (i |f +/.) H,=£ sin * (J f +/.) (34) 

S a =--COS«,-^ ff f =_ ?c08 „_. (35) 

As already stated, the electric and magnetic fields in the air at the boundary can be de- 
termined from the fields in the sea and the boundary conditions. The fields are related through 
the boundary conditions as follows, for z r =0: 

H i2 =H n i=r, <p, z (36a) 

E i2 =E n i=r,<p (36b) 

E z2 = -jgE zl . (36c) 

The boundary conditions (36) state that the magnetic fields and the tangential electric fields 
in air are equal, immediately adjacent to the boundary, to those in the sea, and that the ver- 
tical electric fields in the air are related to those in the sea by the proportionality constant —jg. 

4. Evaluation of the Integrals 

In the previous section the electric and magnetic fields in the sea and at the boundary 
were written as functions of the integrals I a (24) and I b (25). Apparently these integrals can- 
not be evaluated in closed form. Asymptotic expansions for I a and I b (as well as I aU (15a), 
I bh (15b), I a2 (22a), I b2 (22b)) and their appropriate derivatives can be determined by the 
method of critical points [Fredericks, 1953; Banos, 1953, 1954; Erdelyi, 1956]. 

However, it is possible to reduce the integrals I a and I b to those evaluated by Sommerfeld 
[1909, 1926, 1949] for the case in which the source and point of observation both lie on the 
boundary. Sommerfeld evaluated these integrals by transforming them into integrals along 
a contour in the complex plane. It is necessary to investigate the contours used to show the 
approximation possible for the case of the source and point of observation in the sea. 

An integrand map is shown in figure 2. There are four branch points and two poles. 
The branch points are associated with both integrals, but I b does not have a pole. The branch 
points arise because of the presence of the square roots in L and M. They occur where 

L =^¥+J9=0 (37a) 

M=s[i^l -0. (37b) 

B.P. 1 and B.P. 3 are associated with X, and B.P. 2 and B.P. 4 are associated with M. 
Coordinates of B.P. 1 and B.P. 3 are given by 

B.P. 1 *=VP"(-1+J) (38a) 

B.P. 3 f=jgj2(l-j). (38b) 

It can be seen that their distance from the origin is -\'g. For low frequencies and ;high con- 
ductivity, as with sea water, this is a rather large value (generally > 10 3 for communication 
in the sea). 

In the form shown in (37b) it would appear that the branch points 2 and 4 would be on 
the real axis. It is obvious from the limits of the integrals that the path of integration must 


B. P. 1 


i// PLANE 


POLE 1 \ 

B. P. 2 \ 


X/2|\\ i 

X /2 1 I/ 2 a 

i X 

i i i 


t -' 


1/2 g 

\ t X— 

\ A 

\B.P.4 \ 



Figuke 2. Integrand map. 

lie along the real axis and this could cause certain complications. The reason this branch 
point appears to lie on the real axis is that medium 2 has been assumed to have zero conduc- 
tivity. As a matter of fact, any practical medium, even air, has some conductivity and there- 
fore the effective (complex) permittivity must contain a small imaginary component. 

Tims, the complex permittivity, e, is 

€=€ (l-ix) 


where x is a very small value associated with the conductivity of air. Utilizing the complex 
permittivity in (37b) we find the branch points are located by 

B.P. 2 *«-l+jx/2 
B.P. 4 x/s^l-jx/2. 
The poles occur where (L—jgM) is zero. This occurs when 






For the case of large g (generally > 10 6 ) the location of the poles is given approximately by 

Pole 1 f~-l+j/2g 
Pole 2 f~l-j/2g. 


The integrals as originally stated have a lower limit of zero and an infinite upper limit. 
To aid in evaluating them by contour methods it is convenient to write these integrals in 
forms such that the path of integration lies along the entire real axis. The principal conse- 
quence is the conversion of the Bessel functions of the first kind into Hankel functions of the 
first kind. Thus the integrals become 

J 00 
- o 




Hy( P i)+d+. 


It is therefore possible to close a contour by swinging a semicircle, whose radius is un- 
bounded, from the positive real axis to the negative real axis through the upper half plane. 
Since there are two branch points in the upper half plane, one must form the contour such 
that the two associated branch cuts are not crossed. It is, of course, possible to choose con- 
venient branch cuts, within limits. 

Branch line 2 lies along the contour from \p= -l+jx/2 to xp-^j™ for which M is pure 
imaginary. It extends, in essence along the horizontal axis from —1 to and then up along 
the imaginary axis. The contour of integration is shown in figure 3, where it can be seen 
that branch line 2 is the first branch cut encountered in swinging the infinite semicircle from 
positive real axis to negative real axis. The choice of values for the square root in M on the 
two Riemann surfaces separated by branch line 2 is such that for £ r >0, the exponent in the 
factor £** has a positive real part in the first quadrant of the ^-plane and a negative real 
part in the second quadrant. (This aids in obtaining convergence of the integrals in the non- 
conducting medium (22) along branch line 1.) One observes as Sommerfeld did that in this 
case (high conductivity) Pole 1 and B.P. 2 are very close together so that the contours for 
I a must pass around the branch point and the pole with the same loop. This problem has been 
treated in great detail by numerous authors and will not be elaborated here. 

Branch line 1 extends from \[/=-}Jg/2 (— 1+i) to ^— ^'°°. Along this line L has a large 
imaginary component and a small real component. The choice of Riemann surfaces is such 
that the real part of the exponent in the factor e~ L ? is negative on the left side of the branch 
line, because this aids in exponential reduction of the integrand along branch line 2. 

It can be shown readily that the integral along the infinite semicircle is zero. Following 
the terminology used by other writers in the past, the integral along branch line 2 is called 
P +Q2 (except for I b where it is just Q 2 ). The integral along branch line 1 is called Q lt Thus 
for each of the integrals we may state that the value is given by 

I=P+Qi + Q 2 (45) 

where P is, of course, zero for I b . 

For a highly conducting medium, except very close to the source dipole, the contribution 
to the integral along branch line 1 is negligible. One can validate this statement by examining 
the asymptotic expansion of the zero order Hankel function of the first kind [Stratton, 1940] 
which is 


g^(p^ — t/4) 



It is seen that the Hankel function vanishes exponentially as \f/ increases along branch line 1. 
Since the smallest imaginary part of \f/ associated with branch line 1 is -Jg/2, the contribution 
Qi is justifiably neglected. Consequently, it can be assumed that the significant contribution 
to the integral comes from the integration along branch line 2; that is 

/«P+<? 2 . 


v// PLANE 

Figure 3. Contour oj integration. 

The most important approximation made in this treatment is the one allowing the removal 
of the exponential e~ LK from the integrand. To see that this is possible, it is necessary to express 
/, in the form of either (43) or (44), as the sum of two integrals for integration along branch 
line 2. The limits of the first integral are B.P. 2 and a point \f/ on branch line 2; and the limits of 
the second integral arc \p {) and j*> . The point \p is chosen such that l<<|^o| 2 <<i/- 
For examp le, using g=U)\ one might choose ^ =yi0 3 . Consequently, one can approximate 
^ = V^ 2 +i#~ -yljg over the first integral (B.P. 2, ^ ) with negligible error introduced in the 
evaluation of the first integral. Over the second integral (^oJ°°)> the inequality 

\e-^\<\e- L !\ 

permits the following inequality to be written: 













= B 



where a=< 

\ 1 for h 

Using the approximation M=V^— 1 ~*P, over (^ ,i°°), a»d (46) with 10- 2 <p<10 2 ,|i/'|> |ifo>|>10 3 , 
the following upper bound is written for the inequality (49): 


ji r 

V "7> Jif 

if/+(l-a 2 )^ 2 | 


It is shown later that a contribution whose magnitude is less than the upper bound (50) is 
negligible. Consequently, for the following approximation in /: 



one observes that the error is negligible in the first integral and that the contribution of the 
second integral, where the approximation is relatively poor, is also negligible. Hence the 
approximation is justified. Using (51), the integrals I a and I b for the conductivity and fre- 
quency range considered in this treatment are written as follows: 

\I a ~P+Q* -&<->-•» V^r p«" ^ W 




/ 6 «Q 2 «2e ( - 1 -^ ) V«/2f 






This approximation is important because it places outside the integral the factor associated 
with the distances of the dipole and of the point of observation from the boundary. Thus 
the remaining integral is simply that for observation at the surface of fields due to a source 
also located at the surface. This was the problem Sommerfeld [1909] treated for the vertical 
dipole, and the horizontal dipole problem has also been treated since then by many writers. 
Thus the problem for fields due to a dipole in a conducting half space has been reduced to the known 
problem for -fields due to a dipole at an interface. 

Both convergent and asymptotic series were developed by Sommerfeld and others for 
this integral. The expression for I a was found [Moore, 1951] for 0<p<l, to be 


,-W"/-, l,,l j, „, „ , 1 , r . , (J,)' 

4-#^Hr(-p)+57Ar2^i U) (-p)+- • •}• (54) 

This may be expressed as 

I a ~j2 e —^—T. (55) 

For distances such that 10 2 <p<l, conductivity <r~4 mhos/m, and frequency /<^50 kc/s, 
then T~l. A more convenient form for I a given by Sommerfeld [1949] is the following for 

J.=j2 14-| V2x^ e-^-Tje-"' 2 e-" 2/2 ^ • 


An expression for 7 6 similar to that for I a (54) was also obtained by Moore. However, I b 
can be expressed in closed form [Wait, 1952] as follows: 

7 »= 2 ^-iv k+i'-G+VwO'-^-'H (57) 

For the practical purpose of submarine communications; that is, <j~4 mhos/m, /< 50 kc/s, 
it is sufficient to approximate I a (54) or (56) and I b (57) as follows: 

I a =j2 e ~ 3P ~ 39 \ 10~ 2 <P<10 2 , P »r (58) 


p -29— -JTq$ 
J,= -j2 3 (1+j p), 10- 2 < P <1, P »f (59a) 



/*=2^^(l-y, 1<p<10 2 , P »f. (59b) 

Now that the integrals I a and I b have been evaluated, the derivatives of I a and / & involved 
in the field components ((26) through (36)) must be determined. It can be shown [Moore 1951] 
that the power series (54) can be differentiated, allowing the necessary derivatives to the same 
order as of (58) to exist. I b (57) can be differentiated; consequently, the necessary derivatives 
of (59) also exist. Keturning to the statement that ^can be neglected in (16) and (17), one 
can use (16), (17), (24), (25), (58), and (59) to verify this approximation. Similarly, the 
approximation (51) can be justified by comparing the magnitude of the upper bound (50) to 
(58) or (59). For the case #=10 7 , ^ =\7'10 3 and 10~ 2 <p<10 2 , the error is less than 1 percent. 

5. Fields in a Conducting Half Space 

After performing the indicated differentiations, substituting numerical values for per- 
mittivity, permeability, and velocity of propagation in free space, and inserting dipole moments, 
the resulting far field expressions (26) through (35) for p>l, are 

E? «-Vfl-76X10- 82 "^r^ 5 T (l-^) (60) 

H%< ^-lMXlO-" ^ 116 ''" t(i-£) (61) 

o-P \ P/ 

— (l-— ) (62) 


gy « J=j 1 .76X 10- 32 0>V2IS 1/ ?t "* (l- 3 —) (63) 

u p \ p / 

HY « - Vjl.05X10- a ""J,!*,""*' p^-|[] sin » ( 64 ) 

#£*«V-J5.26X10- 2 *^4J! T i_X( 2 r-.l)-4 cos*. (65) 

<f P L. -f p P J 

E** ~-5.92X 10- 27 <B ' 77e ~' / ' T [l-Jr (2T-l)--7\ cos y (66) 

<rp L i p p'J 

^r-Jl-18X10- 26 ^^[^-g] S m^ (67) 

ff »" «- Vj 6.64X10- 30 0>miS f e ~ !n T (\-i-±>i sin v (68) 

o- p \ p p i / 

^r^-V-i 1-33X10- 29 , /2 2 Wl-^lcos* (69) 

"■ p \ p/ 

H'r~jl.l7XlO-™ I -T[1-*-)cqs v (70) 

H*"«-5.83X 10- 27 T ( l-^~ ) sin v . (71) 

p \ p p / 

Similarly, the resulting near field expressions (p<0) are 

Ef « V=7 1 -76 X 10- 32 w J;^ 2 (I+JP) (72) 

gy «jl.57X 10- 29 ^y " (!+J» (73) 

tJISNe-* 1 *' / d 2 \ 

^r«-5.90X10- 35 ^^ ^l+jp-|j (74) 

Hr«-V^5.26X10- 32 ^^p- S (l+ip-|) (75) 

Hf « 7=7 1 .05 X 10- 23 "''^fy ''' (1 +JP) sin ? (76) 

gy « - V=J 5.26X 10- 24 ^fy" (1+JP-P 2 ) cos „ (77) 

# r »«~5.92X10- 27 " 3/ ^ 3 " 2/S (1+ip-P 2 ) cos <p (78) 


^/^1.18X10- 26 t ° 3/ ^r /8 (1+JP) sin *> (79) 

E» m ~4] 6.64X10- 30 " J *f* (1+ip-p 2 ) sin v (80) 

Er~-^3 1.33X10- 29 C ° ^T/y (1+ip) cos ^ (81) 




H r hm ~l.l7XlQ- 26 3 (1+jp) cos ^ (82) 

ff/ w «5.84X10- 27 i0i 3 (1+jp-p 2 ) sin *>. (83) 

Here the superscripts ve, vm, he, and Am refer respectively to vertical-electric, vertical-mag- 
netic, horizontal-electric, and horizontal-magnetic dipoles. T is determined from (54) and 
(55). ^ 

The fields are all expressed in terms of p, the radial distance in units of (free-space-wave- 
length) /2t. Since p is directly proportional to both co and r, the frequency dependence of the 
result is not what it would be if p were independent of frequenc}^. For example, at values of 
10~ 2 <p<O, several of the fields are independent of frequency except for the depth attenuation 
factor: E r he , E/% H r hm , and H p hm . 

The z-components of the fields in the conducting halfspace (sea) are all small (zero to a 
first-order approximation for the vertical electric dipole) compared with the horizontal compo- 
nents, and therefore they have not been included. On the other hand, the ^-component of the 
electric field in the air, for the horizontal dipoles, is the predominant component. The ex- 
pressions for the electric field vertical component in the nonconducting region at the surface 
can be obtained by applying the boundary condition (36c) for p>l; they are 

E\l~-4=3 1.98X10- 21 "* T (\- J -) cos <p (84) 

o?ISNe~ z t' b 
£ , ^^2.22X10- 24 T 

Yl-Asin^. (85) 

Similarly, the results for p< 1 are 

E h A~-Jj 1.98X10- 21 ^— (1+jp) coscp (86) 

o p 

u 3 ISNe~ z t ^ 

ff^-j2.22X10~ 24 2 (1+Jp) sin <p. (87) 


These fields are given only at the surface as no effort has been made to evaluate the necessary 
field integrals at any distance away from the surface in the nonconducting medium. The 
horizontal electric antenna is the most practical for use in submarine communications. As an 
illustration of the manner in which the electric and magnetic fields vary as a function of distance 
from the source; the fields for the horizontal electric antenna have been plotted as a function 
of p in figures 4, 5, and 6, for typical values of frequency, conductivity, and practical antenna 
current, length, and depth. 

6. Discussion and Conclusions 

The form taken by the field equations indicates the mode of propagation that prevails as 
long as the distance from the boundary to both point of observation and source dipole is small 
compared with the horizontal separation. Each of the field expressions may be considered to 
consist of 3 parts: a multiplying factor which includes the dipole strength and parameters such 
as frequency and conductivity of the medium; an exponential attenuation factor whose expo- 
nent is the sum of the distance from the dipole to the surface and from the surface to the point 
of observation, expressed in units of skin depth; and a factor associated with variation in the 
radial (horizontal) direction. The latter factor is the same as that for surface fields of surface 

Thus it appears that propagation occurs as indicated in figure 7. The wave starts at the 
dipole, proceeds by the shortest path to the surface (the path of minimum attenuation), is 
refracted at the surface, travels along the surface as a wave in air, and comes to the point of 
observation by the path of least attenuation (straight down) . 


10 10* 


Figure 4. Variation of the maximum absolute value of the 
magnetic field components in the sea for a submerged hori- 
zontal electric dipole with radial distance in units of free- 
space-wavelength I '2tt. 

10 10* 


Figure 5. Variation of the maximum absolute value of the 
electric field components in the sea for a submerged horizontal 
dipole with radial distance in units of free-space-wavelength/ '2* . 

It is possible to consider that an equivalent source on the surface sets up the wave in the 
nonconducting medium. This wave then travels as would be expected for such a source. 
Since the sea is not a perfect conductor, there is a slight tilt to the wave front, as indicated 
by the presence of both radial and axial components of the electric field; the radial (hori- 
zontal) component is much smaller than the axial (vertical) component. The Poynting vec- 
tor for such a wave is nearly horizontal, but tilted slightly toward the conducting medium. 
The component of the Poynting vector associated with that part of the wave traveling into the 
conducting medium is small but finite. This wave which travels straight down is that which 
is seen by the observer in the conducting medium. 

A similar situation prevails for waves in an imperfectly conducting waveguide. It is 
customary to calculate the losses in the wall of the waveguide by calculating the power flowing 
into the wall per unit area. In the waveguide this represents a leak of power from the desired 
wave. In this situation, where the observer is actually located in the conductor, the desired 
signal is the same as the "leak" for the waveguide. 

Because of this mode of propagation, it is possible for radiation starting with the dipole 
in a conducting half space to be observed at much greater distances than would be possible if 
the wave were generated in an infinite homogeneous conducting medium. Thus, the exponen- 
tial attenuation of 55 db per wavelength applies in the case of the conducting half space only 
to that part of the path from the dipole to the surface and from the surface to the point of 
observation. The rest of the path undergoes the same sort of attenuation that a wave gener- 
ated by a dipole in air would encounter. 


I0 4 

£ 10 




x: n 



v IeUsI 

1 1 1 


\ (slope: 


FREQUENCY, f =10 kc/s 
CONDUCTIVITY ,cr = 4 mhos/m 
ANTENNA CURRENT ,1 = 100 amp 

ANTENNA LENGTH, 1 = 100 m 





>v (slope: 



1 l I 


Figure 6. Variation of the maximum absolute value of the 
normal electric field component in the air for a submerged 
horizontal electric dipole with radial distance in units of free- 










Figure 7. Path of propagation. 


I I 1 


I 1 1 








/ z=8 / 



/ / z = 6 / 













i i i 


1 1 1 

5 7 10 


30 40 50 

Figure 8. Variation of depth attenuation factor with frequency. 

Although this development has applied to the case of a conducting half space and the 
expressions given are those associated with the approximation of the earth's surface by a 
plane, the fact that it is possible to separate out the horizontal attenuation of the wave in air 
means that it is also possible to apply the transmission loss that would be developed using 
spherical, rather than plane, earth geometry. In fact, it is also possible to use the attenuation 
calculated when the ionosphere is taken into account. Thus the results obtained here are 
considerably more general, within the limits of the approximations used, than would be indicated 
by the assumption of a plane earth. 

It is interesting to note the magnitude of the attenuation of the wave in going through 
the conducting medium for the special case of sea water. This is shown for several depths as 
a function of frequency in figure 8. The depth indication is in meters. It is obvious that if 
the attenuation is to be kept low, the frequency must be kept low also. 

It has been shown here that the fields created by a dipole in a conducting medium, which 
is near a plane boundary with a nonconducting medium, may be expressed in a form that may 
be considered as propagation from the dipole to the interface, travel through the nonconducting 
medium along the boundary, and propagation back into the conducting medium to the point 
of observation. Expressions have been presented for the fields of both vertical and horizon- 
tal, electric and magnetic dipoles. The fields of the horizontal dipoles are considerably stronger 
than those of the vertical dipoles, as would be expected because of the vertical nulls in the 
radiation from the vertical dipoles. 


7. References 

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Fredricks, K. O., Special topics of analysis, New York Univ. Inst, of Math. Sci. (1953). 
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Sommerfeld, A., Uber die Ausbreitung Elektromagnetischer Wellen uber ein eben Erde, Ann. der Phys. Ser. 4, 

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(Paper 65D6-159)